How a 100-year-old math quirk helps us clean up mri brain scans
We found that noiseless MRIs follow Benford’s Law (the ‘1s’ rule). We use this discovery to build a new machine learning tool that can accurately measure the noise level in brain scans.
Magnetic Resonance Images (MRIs) are incredible tools, but they have a classic problem: Rician noise. This is the grainy “static” that can obscure tiny but critical details in a brain scan. To clean up a noisy MRI, you first need to know how noisy it is. But accurately measuring that noise is a notoriously difficult problem.
In our 2023 paper published in Axioms, we introduced a brand new method to solve this, based on a fascinating mathematical law first observed over 100 years ago.
🧐 The problem: you can’t fix a problem you can’t measure
When an MRI scan is noisy, fine details of brain structures can be lost. This makes diagnosis harder. While we have many algorithms to denoise an image, their performance depends heavily on knowing the exact level of noise to begin with.
If you estimate the noise too low, you leave the image grainy. If you estimate it too high, you blur away important details along with the noise. The challenge is getting a fast, accurate, and reliable noise “reading.”
💡 Our solution: a strange, 100-year-old math law
Our solution comes from a counter-intuitive observation known as Benford’s Law. This law states that in many natural datasets (stock prices, river lengths, populations), the number ‘1’ appears as the first significant digit about 30% of the time, while ‘9’ appears less than 5% of the time.
Our “Aha!” moment was when we applied this to MRI scans. We discovered:
- If you take a clean, noiseless T1 MRI and look at its frequency coefficients (the data behind the image), the first digits perfectly follow Benford’s Law.
- Crucially, Rician noise breaks this law. The noise messes up this natural distribution, making the digits more random.
This gives us a new way to measure noise: just measure how broken Benford’s Law is.
🛠️ How it works: measuring the “broken-ness”
If a clean scan follows the law and a noisy scan deviates from it, then the amount of deviation should be directly proportional to the amount of noise.
We built a system that: 1. Takes an MRI scan and calculates its frequency transforms (Fourier, Cosine, etc.). 2. Measures the distribution of the first significant digits of these coefficients. 3. Uses statistical distance metrics (like Kullback–Leibler divergence) to get a single number for “how far” this distribution is from the ideal Benford’s Law. 4. Feeds this “deviation score” into a supervised machine learning model (a regressor) that learns to map it to a precise noise level.
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🚀 The results: a new, reliable noise-meter
It worked. We validated our method on hundreds of MRI scans from several different datasets and machines (both 1.5T and 3T).
The results showed that our new, logic-based method was highly competitive with, and in many cases even surpassed, other established, state-of-the-art noise estimation techniques.
🔬 Why does this matter?
This isn’t just a math curiosity. It’s a new, practical, and reliable tool for medical imaging.
Having a fast and accurate noise estimator is the critical first step for any advanced denoising algorithm. By providing a better way to measure the noise, we pave the way for better ways to remove it. This leads to clearer, more trustworthy MRI scans, which ultimately helps doctors make more accurate diagnoses.
📖 The full paper
For the complete technical breakdown, the statistical analysis, and the full experimental results across all datasets, you can read the original open-access article in Axioms.
Regression of the Rician Noise Level in 3D Magnetic Resonance Images from the Distribution of the First Significant Digit. Authors: Rosa Maza-Quiroga, Karl Thurnhofer-Hemsi, Domingo López-Rodríguez, Ezequiel López-Rubio. Journal: Axioms (vol. 12, issue 12, 1117)